DSolve for System of Non-Homogeneous Differential Equation

Question

Solve: x' = Ax + b Given: A = {{1, 2}, {3, 4}}; b = {{e^{t}},{2}};

This will solve if b == 0

DSolve[{x1'[t], x2'[t]} == a.{x1[t], x2[t]}, {x1[t], x2[t]}, t]

But if I add "b" I get "DSolve::nolist: ...there should be no lists on either side of the equation" error.

DSolve[{x1'[t], x2'[t]} == a.{x1[t], x2[t]}+b, {x1[t], x2[t]}, t]

Could it be that your variable name is a capital A and you DSolve code has a lower case a? For instance DSolve[{x1'[t], x2'[t]} == A.{x1[t], x2[t]} + b, {x1[t], x2[t]}, t] works for me. — Michael E2, Commented Oct 11, 2020 at 14:01
Also, generally I avoid starting variable names with capitals so as not to conflict with built-in functions and variables. See #4 in mathematica.stackexchange.com/a/18395/4999 — Michael E2, Commented Oct 11, 2020 at 14:02

Cesareo · Accepted Answer · 2020-10-11 15:03:59Z

1

Try this

A = {{1, 2}, {3, 4}};
b = {E^t, 2};
X = {x1[t], x2[t]};
cinits = Thread[(X /. {t -> 0}) == {1, 2}];
DSolve[{D[X, t] == A.X + b, cinits}, X, t]

answered Oct 10, 2020 at 23:24

Cesareo

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$\begingroup$ Why does this work as opposed to what I have? And how would you add initial condition into the DSolve[Thread that you have? Say we have x0[t] = 1 and x1[t] = 2? Thanks! $\endgroup$
– Mathhelp
Commented Oct 11, 2020 at 5:15
$\begingroup$ There are some syntax rules regarding lists handling... $\endgroup$
– Cesareo
Commented Oct 11, 2020 at 8:17
$\begingroup$ @Mathhelp Note that DSolve[{D[X, t] == A.X + b, cinits}, X, t] works without Thread. $\endgroup$
– Michael E2
Commented Oct 11, 2020 at 13:52
$\begingroup$ Yes. It is now fixed. Thanks. $\endgroup$
– Cesareo
Commented Oct 11, 2020 at 15:04

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